Ordinary and spectral extremal problems on vertex disjoint copies of even fans

TL;DR

This paper investigates the maximum size and spectral radius of graphs that do not contain multiple disjoint copies of even fan graphs, providing characterizations of extremal graphs for large n.

Contribution

It extends extremal graph theory to multiple disjoint even fan graphs, offering new bounds and characterizations for large graphs.

Findings

Determined extremal numbers for multiple disjoint even fans.

Characterized extremal graphs for large n.

Provided bounds on spectral radius for these graphs.

Abstract

Let $\mathrm{ex}(n, F)$ and $\mathrm{spex}(n, F)$ be the maximum size and spectral radius among all $F$-free graphs with fixed order $n$, respectively. A fan is a graph $P_1\vee P_{s}$ (join of a vertex and a path of order $s$) for $s\ge 3$, and it is called an even fan if $s$ is even. In this paper, we study $\mathrm{ex}(n,t(P_1\vee P_{2k}))$, $\mathrm{spex}(n,t(P_1\vee P_{2k}))$ with $t\ge 1$ and $k\ge 3$ and characterize the corresponding extremal graphs for sufficiently large $n$.